TsT Transformations Research Articles

Homogeneous Yang–Baxter deformations as generalized diffeomorphisms

Yang–Baxter (YB) deformations of string sigma model provide deformed target spaces. We propose that homogeneous YB deformations always lead to a certain class of β-twisted backgrounds and represent the bosonic part of the supergravity fields in terms of the classical r-matrix associated with the YB deformation. We then show that various β-twisted backgrounds can be realized by considering generalized diffeomorphisms in the undeformed background. Our result extends the notable relation between the YB deformations and (non-commuting) TsT transformations. We also discuss more general deformations beyond the YB deformations.

Abelian Yang–Baxter deformations and TsT transformations

We prove that abelian Yang–Baxter deformations of superstring coset σ models are equivalent to sequences of commuting TsT transformations, meaning T dualities and coordinate shifts. Our results extend also to fermionic deformations and fermionic T duality, and naturally lead to a TsT subgroup of the superduality group OSp(db,db|2df). In cases like AdS5×S5, fermionic deformations necessarily lead to complex models. As an illustration of inequivalent deformations, we give all six abelian deformations of AdS3. We comment on the possible dual field theory interpretation of these (super-)TsT models.

Almost abelian twists and AdS/CFT

A large class of the recently found unimodular nonabelian homogeneous Yang-Baxter deformations of the AdS_5 x S^5 superstring can be realized as sequences of noncommuting TsT transformations. I show that many of them are duals to various noncommutative versions of supersymmetric Yang-Mills theory, structurally determined directly in terms of the associated r matrices, in line with previous expectations in the literature.

Homogeneous Yang–Baxter deformations as non-abelian duals of the AdS5 σ-model

We propose that the Yang–Baxter deformation of the symmetric space σ-model parameterized by an r-matrix solving the homogeneous (classical) Yang–Baxter equation is equivalent to the non-abelian dual of the undeformed model with respect to a subgroup determined by the structure of the r-matrix. We explicitly demonstrate this on numerous examples in the case of the σ-model. The same should also be true for the full supercoset model, providing an explanation for and generalizing several recent observations relating homogeneous Yang–Baxter deformations based on non-abelian r-matrices to the undeformed model by a combination of T-dualities and nonlinear coordinate redefinitions. This also includes the special case of deformations based on abelian r-matrices, which correspond to TsT transformations: they are equivalent to non-abelian duals of the original model with respect to a central extension of abelian subalgebras.

On classical Yang-Baxter based deformations of the AdS5 × S5 superstring

Interesting deformations of AdS5 × S5 such as the gravity dual of noncommutative SYM and Schödinger spacetimes have recently been shown to be integrable. We clarify questions regarding the reality and integrability properties of the associated construction based on R matrices that solve the classical Yang-Baxter equation, and present an overview of manifestly real R matrices associated to the various deformations. We also discuss when these R matrices should correspond to TsT transformations, which not all do, and briefly analyze the symmetries preserved by these deformations, for example finding Schrödinger superalgebras that were previously obtained as subalgebras of $$ \mathfrak{p}\mathfrak{s}\mathfrak{u} $$ (2, 2|4). Our results contain a (singular) generalization of an apparently non-TsT deformation of AdS5 × S5, whose status as a string background is an interesting open question.

Schrödinger geometries arising from Yang-Baxter deformations

We present further examples of the correspondence between solutions of type IIB supergravity and classical $r$-matrices satisfying the classical Yang-Baxter equation (CYBE). In the previous works, classical $r$-matrices have been composed of generators of only one of either $\mathfrak{so}(2,4)$ or $\mathfrak{so}(6)$. In this paper, we consider some examples of $r$-matrices with both of them. The $r$-matrices of this kind contain (generalized) Schr\"odinger spacetimes and gravity duals of dipole theories. It is known that the generalized Schr\"odinger spacetimes can also be obtained via a certain class of TsT transformations called null Melvin twists. The metric and NS-NS two-form are reproduced by following the Yang-Baxter sigma-model description.

Yang-Baxter deformations and string dualities

We further study integrable deformations of the AdS$_5\times$S$^5$ superstring by following the Yang-Baxter sigma model approach with classical $r$-matrices satisfying the classical Yang-Baxter equation (CYBE). Deformed string backgrounds specified by $r$-matrices are considered as solutions of type IIB supergravity, and therefore the relation between gravitational solutions and $r$-matrices may be called the gravity/CYBE correspondence. In this paper, we present a family of string backgrounds associated with a classical $r$-matrices carrying two parameters and its three-parameter generalization. The two-parameter case leads to the metric and NS-NS two-form of a solution found by Hubeny-Rangamani-Ross [hep-th/0504034] and another solution in [arXiv:1402.6147]. For all of the backgrounds associated with the three-parameter case, the metric and NS-NS two-form are reproduced by performing TsT transformations and S-dualities for the undeformed AdS$_5\times$S$^5$ background. As a result, one can anticipate the R-R sector that should be reproduced via a supercoset construction.

Deformations of T 1,1 as Yang-Baxter sigma models

We consider a family of deformations of T 1,1 in the Yang-Baxter sigma model approach. We first discuss a supercoset description of T 1,1, which makes manifest the full symmetry of the space and leads to the standard Sasaki-Einstein metric. Next, we consider three-parameter deformations of T 1,1 by using classical r-matrices satisfying the classical Yang-Baxter equation (CYBE). The resulting metric and NS-NS two-form agree exactly with the ones obtained via TsT transformations, and contain the Lunin-Maldacena background as a special case. It is worth noting that for AdS5 × T 1,1 , classical integrability for the full sector has been argued to be lost. Hence our result indicates that the Yang-Baxter sigma model approach is applicable even for non-integrable cosets. This observation suggests that the gravity/CYBE correspondence can be extended beyond integrable cases.

Asymptotically Schrödinger space‐times

AbstractWe first review how asymptotically Schrödinger space‐times arise in a natural way by performing a TsT transformation on asymptotically AdS space‐times and some of its consequences. We then give a coordinate independent definition of a pure Schrödinger space‐time in terms of an AdS metric and an AdS null Killing vector. Furthermore, by analogy with the AdS case, we provide a local coordinate independent definition of a Schrödinger boundary in terms of a defining function. Finally this directly leads us to the construction of a Fefferman‐Graham expansion for locally Schrödinger space‐times.

The geometry and phase structure of non-relativistic branes

We use the solution-generating technique, the TsT transformation, to obtain new solutions in type II string theory as well as in M-theory. We explicitly work out examples starting with rotating D3 and M2 branes as well as the D1–D5-p system. Among a variety of solutions, we find many of them having asymptotic Schrödinger symmetry. We also devise a new method of deriving the free energy of black brane systems, which is more efficient than the Euclidean action procedure. We test our method on known examples before applying it to new asymptotically Schrödinger backgrounds. We study the phase structure of these backgrounds by analyzing the free energy thus derived.

Non-relativistic metrics with extremal limits

We present solutions of type IIB supergravity with z = 2 Schrodinger asymptotics that admit an extremal limit, i.e. the black hole horizon has a double zero. These solutions are obtained as TsT transformations of the charged planar black hole in AdS 5 × S 5. Unlike the uncharged solution, the Ramond-Ramond two-form is turned on. We study the thermodynamic properties of these new solutions, and we show that the ratio of shear viscosity to entropy density is 1/4π even in the extremal limit. We also consider the TsT-transformed soliton and show that, for a special radius of the compact circle, there is a confinement-deconfinement phase transition at zero temperature between the soliton and black hole phases. © SISSA 2009.

Phase structure ofβ-deformedN=4super Yang-Mills theory onS3with chemical potentials

We study the {beta} deformation of N=4 super Yang-Mills theory on S{sup 3} with chemical potentials for the U(1){sub R} as well as the two global U(1) symmetries. The one-loop effective potential at weak coupling is computed for both the Coulomb and Higgs branches. At near critical chemical potential and small finite temperature, we find a metastable state at the origin of moduli space. On the Higgs branch, this has the interpretation in terms of deconstruction as an extra-dimensional torus which becomes metastable for infinite size and decays to zero size through quantum tunneling and thermal activation. At strong coupling, the theory is described by its gravitational dual. The relevant background is found by performing a TsT transformation on the solution describing an AdS{sub 5} black hole spinning in S{sup 5}. A probe-brane calculation, using giant gravitons as probes, reveals qualitative agreement with the weak coupling results.

Classical integrability of strings in γ-deformed backgrounds

This paper considers classical strings propagating in γ-deformed AdS3 × S3 backgrounds generated by certain shift T-dualities accompanied (TsT) transformations on S3 and AdS3, respectively. It finds that the U(1) currents of strings with the twisted boundary conditions are equal to those in γ-deformed backgrounds generated by TsT transformations on both S3 and AdS3. Applying the TsT transformations, it derives the local Lax connections and the monodromy matrices in γ-deformed backgrounds with the spectral parameter which ensure the classical integrability of the string theories.

On deformed gauge theories and their string/M-theory duals

We present general formulae for the TsT transformation (T-duality, shift, T-duality) of type II string backgrounds and open string boundary conditions. The TsT transformation provides a systematic procedure to find string theory duals of gauge theories with deformed products of fields in the lagrangian, and the duals can be analyzed by using transformed D-brane probes. As examples illustrating some features of the deformed theories, we consider the known backgrounds dual to non-commutative, dipole and β-deformed N = 4 Super Yang-Mills as well as new backgrounds dual to deformations of the recently proposed N = 6 Chern-Simons-matter theory living on multiple M2-branes on an orbifold. © 2008 SISSA.

Integrability and Gauge Fixing for Bosonic String on AdS3 × S3

The integrability and the TST transformation of the bosonic string on AdS3 × S3 are studied. The Lax connection for gauge fixed theory is constructed and proved to be flat.

Giant magnons in TsT-transformed AdS5× S5

We consider giant magnons propagating in a ?-deformed AdS5 ? S5 background obtained from AdS5 ? S5 by means of a chain of TsT transformations. We point out that in the light-cone gauge and in the infinite J limit the deformed and undeformed string models share the same magnon dispersion relation, the (2|2)???(2|2)-invariant world-sheet S-matrix and the dressing factor. The ?-dependence in the limit is only due to different level-matching conditions. We consider the reduction of the deformed model to R ? S3 and determine the leading ?-dependence of the dispersion relation for a finite J giant magnon.

Lax Connection of Strings in γ-Deformed Backgrounds

In this paper we consider classical strings propagating in γ-deformed AdS3 × S3 backgrounds generated by TsT transformation on the AdS3 sector, which is described as the group manifold SL(2,R), then we prove that the U(1) currents of strings with the twisted boundary conditions are equal to those in γ-deformed backgrounds. Using TsT transformation, we can derive the local Lax connection and the monodromy matrix in γ-deformed backgrounds with the spectral parameter, which ensures the classical integrability of the string theories.

Pure spinor strings in TsT deformed background

We consider pure spinor strings that propagate in the background generated by a sequence of TsT transformations. We use the fact that U(1) isometry variables of TsT-transformed background are related to the isometry variables of the initial background in the universal way that is independent of the details of the background. We will argue that after redefinitions of pure spinors and the fermionic variables we can construct pure spinor action with manifest U(1) isometry. This fact implies that the pure spinor string in TsT-transformed background is described by pure spinor string in the original background where world-volume modes are subject to twisted boundary conditions. We will argue that these twisted boundary conditions generally prevent to prove the quantum conformal invariance of the pure spinor string in AdS_5 x S^5 background. We determine the conditions under which this quantum conformal invariance can be proved. We also determine the Lax pair for pure spinor strings in the TsT-transformed background.

Green-Schwarz Strings in TsT-transformed backgrounds

We consider classical strings propagating in a background generated by a sequence of TsT transformations. We describe a general procedure to derive the Green-Schwarz action for strings. We show that the U(1) isometry variables of the TsT-transformed background are related to the isometry variables of the initial background in a universal way independent of the details of the background. This allows us to prove that strings in the TsT-transformed background are described by the Green-Schwarz action for strings in the initial background subject to twisted boundary conditions. Our construction implies that a TsT transformation preserves integrability properties of the string sigma model. We discuss in detail type IIB strings propagating in the \g_i-deformed AdS_5 x S^5 space-time, find the twisted boundary conditions for bosons and fermions, and use them to write down an explicit expression for the monodromy matrix. We also discuss string zero modes whose dynamics is governed by a fermionicgeneralization of the integrable Neumann model.

String spectrum of curved string backgrounds obtained by T-duality and shifts of polar angles

A class of exactly solvable string models can be obtained by starting with flat space and combining T-duality and shifts of angular coordinates of several polar planes. The models are the analog of the Lunin-Maldacena \beta-deformation of the AdS_5 x S^5 type IIB string background, which is dual to a Leigh-Strassler deformation of \N=4 Super Yang-Mills Theory. We determine the complete physical string spectrum for two string models obtained in this way, by explicitly solving the string equations and quantizing in terms of free creation and annihilation operators. We also show that the 3-parameter (b_1,b_2,b_3) model, obtained by three independent TsT transformations, has tachyons in some regions of the parameter space.